This invention relates to apparatus and method for compensating for atmospheric turbulence, and, more particularly, to the use of sparse matrix algebra to determine in a deformable mirror having several actuators which actuators to move and how much in order to compensate for atmospheric turbulence.
Light traveling through the air undergoes distortions from several sources, including turbulence, wind, water vapor and heat, which cause rapid, small-scale variations in the air's density and hence its refractive index. Consequently, the optical characteristics of the air are changing continuously, both spatially and temporally.
Adaptive optics systems have been employed to correct for distortions of light waves traveling through the atmosphere. Typical adaptive optics systems work as a feed back loop in that a sensor detects a beam distortion and sends a signal to a controller to adjust the optics to remove the distortion.
In one type of system, a deformable mirror, provided with a number of actuators which each can independently deform a portion of the mirror, is employed. These actuators move in and out to adjust the mirror's shape and thus replicate the shape of the optical wavefront. It will be appreciated that the movement of the actuators is on the order of fractions of a wavelength of light.
The typical wavefront control system uses a wavefront slope sensor such as a Hartmann sensor, and uses that information to calculate the commands driving the actuators on a deformable mirror, which corrects for the measured wavefront aberrations or distortions.
In such a conventional adaptive optics system, the estimator is the matrix that is multipled by the measurement vector to obtain an estimate of the wavefront distortion. Although the matrix equations required to compute the wavefront estimator are straight forward, the number of computations required for their direct evaluation is becoming unreasonable for some adaptive optics systems.
Starting with the measurement equation Hx=s, the conventional estimate is EQU x=(H.sup.T H).sup.-1 H.sup.T s,
where H represents the actuator-subaperture Jacobian (wavefront sensor derivative), x is the vector of actuator commands representing the wavefront error, s is the wavefront slope vector measured by the wavefront slope sensor, x is the actuator command estimate vector and H.sup.T is the transpose of H.
Direct evaluation of these equations requires the inversion of an n by n matrix (and therefore approximately n.sup.3 /6 adds and multiplies). Calculating the estimate requires n.sup.2 adds and multiplies, and this may also be excessive for systems that require a high temporal bandwidth. The parameter n refers to the number of wavefront distortion degrees of freedom, such as Zernike polynomial coefficients or commands to deformable mirror actuators. In some proposed systems, n may be over 1,000.
A related estimation algorithm first estimates the OPD (optical path difference) and then performs an amplitude-weighted fit of the actuators to the resulting OPD map. Here, let o be the OPD vector. Let Go=s. G is the wavefront sensor/OPD Jacobian, corrected for amplitude effects. Define F so that Fx=o, where F is the OPD/actuator Jacobian. EQU o=(G.sup.T G).sup.-1 G.sup.T s
and EQU x=(F.sup.T W F).sup.-1 F.sup.T Wo.
Here, there will be many more OPD points than slope measurement, to allow for accurate wavefront sensor calibration and weighting of the fit. In an actual implementation of this algorithm, each OPD point corresponds to a pixel on some detector such as a Charge Coupled Device (CCD) or Charge Injection Device (CID). Again, there is a large calculation. The OPD reconstruction will dominate the calculation, requiring the inversion of an n.times.n matrix, where n may reach 10,000 or more. The required n.sup.3 operations would take a long time. Even the n.sup.2 operations required for each estimation, assuming that the estimator has somehow been calculated, are enough to contraindicate this method for large systems despite its superior theoretical performance.
As interest has centered on decreasing the wavelength of light used, the number of actuators for a given deformable mirror must increased. Calculating which actuators must move and the extent of such movement clearly will involve a substantial number of calculations to be performed merely to make one series of adjustments in the deformable mirror. Since the required calculations must be made rather frequently, on the order of 1,000 times per second, it can be seen that an increase in the number of actuators from, say, 100 to 1,000, will tax the computational powers of most affordable computers. Thus, the manner in which the calculations must be performed must be improved.